Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1008,2,Mod(529,1008)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1008.529");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1008.q (of order \(3\), degree \(2\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(8.04892052375\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{3})\) |
Twist minimal: | no (minimal twist has level 504) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
529.1 | 0 | −1.71918 | − | 0.210723i | 0 | 0.0309846 | − | 0.0536670i | 0 | 0.981674 | − | 2.45689i | 0 | 2.91119 | + | 0.724543i | 0 | ||||||||||
529.2 | 0 | −1.64138 | + | 0.553060i | 0 | −0.263002 | + | 0.455533i | 0 | 0.333150 | + | 2.62469i | 0 | 2.38825 | − | 1.81556i | 0 | ||||||||||
529.3 | 0 | −1.04182 | + | 1.38370i | 0 | 1.33425 | − | 2.31099i | 0 | −2.54743 | + | 0.714566i | 0 | −0.829236 | − | 2.88312i | 0 | ||||||||||
529.4 | 0 | −0.987504 | − | 1.42297i | 0 | −1.38590 | + | 2.40045i | 0 | −1.74026 | + | 1.99286i | 0 | −1.04967 | + | 2.81037i | 0 | ||||||||||
529.5 | 0 | −0.748111 | − | 1.56216i | 0 | 2.11148 | − | 3.65719i | 0 | −2.19338 | − | 1.47956i | 0 | −1.88066 | + | 2.33733i | 0 | ||||||||||
529.6 | 0 | 0.633073 | − | 1.61221i | 0 | −1.70368 | + | 2.95086i | 0 | 0.410295 | − | 2.61374i | 0 | −2.19844 | − | 2.04129i | 0 | ||||||||||
529.7 | 0 | 0.704143 | + | 1.58246i | 0 | −1.05220 | + | 1.82246i | 0 | −2.58382 | − | 0.569079i | 0 | −2.00837 | + | 2.22856i | 0 | ||||||||||
529.8 | 0 | 0.914508 | − | 1.47094i | 0 | 0.891774 | − | 1.54460i | 0 | 2.54386 | + | 0.727153i | 0 | −1.32735 | − | 2.69038i | 0 | ||||||||||
529.9 | 0 | 1.46869 | + | 0.918117i | 0 | 1.89970 | − | 3.29038i | 0 | 0.841809 | + | 2.50826i | 0 | 1.31412 | + | 2.69686i | 0 | ||||||||||
529.10 | 0 | 1.69989 | + | 0.332219i | 0 | −1.59750 | + | 2.76695i | 0 | 1.66645 | + | 2.05498i | 0 | 2.77926 | + | 1.12947i | 0 | ||||||||||
529.11 | 0 | 1.71769 | − | 0.222607i | 0 | 0.234085 | − | 0.405446i | 0 | −0.212345 | − | 2.63722i | 0 | 2.90089 | − | 0.764737i | 0 | ||||||||||
625.1 | 0 | −1.71918 | + | 0.210723i | 0 | 0.0309846 | + | 0.0536670i | 0 | 0.981674 | + | 2.45689i | 0 | 2.91119 | − | 0.724543i | 0 | ||||||||||
625.2 | 0 | −1.64138 | − | 0.553060i | 0 | −0.263002 | − | 0.455533i | 0 | 0.333150 | − | 2.62469i | 0 | 2.38825 | + | 1.81556i | 0 | ||||||||||
625.3 | 0 | −1.04182 | − | 1.38370i | 0 | 1.33425 | + | 2.31099i | 0 | −2.54743 | − | 0.714566i | 0 | −0.829236 | + | 2.88312i | 0 | ||||||||||
625.4 | 0 | −0.987504 | + | 1.42297i | 0 | −1.38590 | − | 2.40045i | 0 | −1.74026 | − | 1.99286i | 0 | −1.04967 | − | 2.81037i | 0 | ||||||||||
625.5 | 0 | −0.748111 | + | 1.56216i | 0 | 2.11148 | + | 3.65719i | 0 | −2.19338 | + | 1.47956i | 0 | −1.88066 | − | 2.33733i | 0 | ||||||||||
625.6 | 0 | 0.633073 | + | 1.61221i | 0 | −1.70368 | − | 2.95086i | 0 | 0.410295 | + | 2.61374i | 0 | −2.19844 | + | 2.04129i | 0 | ||||||||||
625.7 | 0 | 0.704143 | − | 1.58246i | 0 | −1.05220 | − | 1.82246i | 0 | −2.58382 | + | 0.569079i | 0 | −2.00837 | − | 2.22856i | 0 | ||||||||||
625.8 | 0 | 0.914508 | + | 1.47094i | 0 | 0.891774 | + | 1.54460i | 0 | 2.54386 | − | 0.727153i | 0 | −1.32735 | + | 2.69038i | 0 | ||||||||||
625.9 | 0 | 1.46869 | − | 0.918117i | 0 | 1.89970 | + | 3.29038i | 0 | 0.841809 | − | 2.50826i | 0 | 1.31412 | − | 2.69686i | 0 | ||||||||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1008.2.q.l | 22 | |
3.b | odd | 2 | 1 | 3024.2.q.l | 22 | ||
4.b | odd | 2 | 1 | 504.2.q.c | ✓ | 22 | |
7.c | even | 3 | 1 | 1008.2.t.l | 22 | ||
9.c | even | 3 | 1 | 1008.2.t.l | 22 | ||
9.d | odd | 6 | 1 | 3024.2.t.k | 22 | ||
12.b | even | 2 | 1 | 1512.2.q.d | 22 | ||
21.h | odd | 6 | 1 | 3024.2.t.k | 22 | ||
28.g | odd | 6 | 1 | 504.2.t.c | yes | 22 | |
36.f | odd | 6 | 1 | 504.2.t.c | yes | 22 | |
36.h | even | 6 | 1 | 1512.2.t.c | 22 | ||
63.h | even | 3 | 1 | inner | 1008.2.q.l | 22 | |
63.j | odd | 6 | 1 | 3024.2.q.l | 22 | ||
84.n | even | 6 | 1 | 1512.2.t.c | 22 | ||
252.u | odd | 6 | 1 | 504.2.q.c | ✓ | 22 | |
252.bb | even | 6 | 1 | 1512.2.q.d | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.q.c | ✓ | 22 | 4.b | odd | 2 | 1 | |
504.2.q.c | ✓ | 22 | 252.u | odd | 6 | 1 | |
504.2.t.c | yes | 22 | 28.g | odd | 6 | 1 | |
504.2.t.c | yes | 22 | 36.f | odd | 6 | 1 | |
1008.2.q.l | 22 | 1.a | even | 1 | 1 | trivial | |
1008.2.q.l | 22 | 63.h | even | 3 | 1 | inner | |
1008.2.t.l | 22 | 7.c | even | 3 | 1 | ||
1008.2.t.l | 22 | 9.c | even | 3 | 1 | ||
1512.2.q.d | 22 | 12.b | even | 2 | 1 | ||
1512.2.q.d | 22 | 252.bb | even | 6 | 1 | ||
1512.2.t.c | 22 | 36.h | even | 6 | 1 | ||
1512.2.t.c | 22 | 84.n | even | 6 | 1 | ||
3024.2.q.l | 22 | 3.b | odd | 2 | 1 | ||
3024.2.q.l | 22 | 63.j | odd | 6 | 1 | ||
3024.2.t.k | 22 | 9.d | odd | 6 | 1 | ||
3024.2.t.k | 22 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):
\( T_{5}^{22} - T_{5}^{21} + 39 T_{5}^{20} - 4 T_{5}^{19} + 952 T_{5}^{18} + 120 T_{5}^{17} + 14145 T_{5}^{16} + \cdots + 5476 \) |
\( T_{11}^{22} + 3 T_{11}^{21} + 80 T_{11}^{20} + 249 T_{11}^{19} + 4059 T_{11}^{18} + 12546 T_{11}^{17} + \cdots + 4241135376 \) |